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Hölder estimates and weak convergences of certain weighted sum processes

Shigeki Aida, Nobuaki Naganuma

TL;DR

The article analyzes weighted sum processes driven by fractional Brownian motion and built from fixed-order Wiener chaos, deriving Hölder-type estimates and a functional limit theorem. By combining Malliavin calculus, the fourth moment theorem, and multidimensional Young integrals, it establishes moment bounds and discrete Hölder control for chaos of order $2$, and proves a weak convergence (FCLT) to Brownian-type stochastic integrals driven by independent Brownian motions. The work further extends these results to Wiener chaos of order $3$, providing a comprehensive covariance analysis across multiple configurations and demonstrating how the limiting behavior can be captured by explicit Gaussian processes. These findings illuminate the asymptotic behavior of weighted Hermite variations and have implications for error analysis in rough differential equations driven by fBm, particularly in multidimensional settings.

Abstract

We study weighted sum processes associated to elements in a Wiener chaos with fixed order. More precisely, we show Hölder estimates and a functional limit theorem for them. Main tools we use are the integration by parts formula in Malliavin calculus, the fourth moment theorem, and estimates in multidimensional Young integrals.

Hölder estimates and weak convergences of certain weighted sum processes

TL;DR

The article analyzes weighted sum processes driven by fractional Brownian motion and built from fixed-order Wiener chaos, deriving Hölder-type estimates and a functional limit theorem. By combining Malliavin calculus, the fourth moment theorem, and multidimensional Young integrals, it establishes moment bounds and discrete Hölder control for chaos of order , and proves a weak convergence (FCLT) to Brownian-type stochastic integrals driven by independent Brownian motions. The work further extends these results to Wiener chaos of order , providing a comprehensive covariance analysis across multiple configurations and demonstrating how the limiting behavior can be captured by explicit Gaussian processes. These findings illuminate the asymptotic behavior of weighted Hermite variations and have implications for error analysis in rough differential equations driven by fBm, particularly in multidimensional settings.

Abstract

We study weighted sum processes associated to elements in a Wiener chaos with fixed order. More precisely, we show Hölder estimates and a functional limit theorem for them. Main tools we use are the integration by parts formula in Malliavin calculus, the fourth moment theorem, and estimates in multidimensional Young integrals.
Paper Structure (16 sections, 33 theorems, 245 equations)

This paper contains 16 sections, 33 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Finite dimensional approximations of iterated integrals
  5. Malliavin derivatives of iterated rough integrals
  6. Moment estimates of weighted sum processes of Wiener chaos of order 2
  7. Weak convergence of (weighted) sum processes of Wiener chaos of order 2
  8. Hölder estimates of (weighted) sum processes of Wiener chaos of order $3$
  9. Proof of Proposition \ref{['estimate of third order']}
  10. Covariance of Wiener chaos of order $3$
  11. The case $K^{m}_{\tau^m_{i-1},\tau^m_i} = B^{\alpha}_{\tau^m_{i-1},\tau^m_i}B^{\beta}_{\tau^m_{i-1},\tau^m_i}B^{\gamma}_{\tau^m_{i-1},\tau^m_i}$
  12. The case $K^{m}_{\tau^m_{i-1},\tau^m_i}=B^{\alpha,\beta}_{\tau^m_{i-1},\tau^m_i}B^{\gamma}_{\tau^m_{i-1},\tau^m_i}$
  13. The case $K^{m}_{\tau^m_{i-1},\tau^m_i}=B^{\alpha,\beta,\gamma}_{\tau^m_{i-1},\tau^m_i}$
  14. Multidimensional Young integral
  15. Definitions and basic results
  16. ...and 1 more sections

Key Result

Theorem 2.2

Let $(F_t)\in \mathcal{J}({\mathbb R})$ and $1\leq \alpha,\beta\leq d$ be distinct. Let $0\le s\le t\le 1$. Let For any positive integer $p$, there exists a positive constant $C_p$ that is independent of $m$ such that

Theorems & Definitions (71)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • ...and 61 more